Probability and Likelihood

Likelihood needed for many of ADMB’s features Standard deviation Variance-covariance matrix Profile likelihood Bayesian MCMC Random effects See Hilborn and Mangel 1997 for a simple introduction See Pawitan 2001 for a comprehensive description

Probability distributions Probability of an event given a probability distribution Probability distribution defined by its form and the values of its parameters

Use of probability distributions Gambling, working out what is the best bet in a game of cards

What we desire The probability of a parameter given the information (data) we have (observed)

Likelihood: compare the probability of the observed data under different values of the parameter The outcome 3 is more probable if the true parameter value is 0.6

Likelihood: a numerical quantity to express the order of preference of values of the parameter MLE

Normal distribution maximum likelihood (one data point) Likelihood -ln(Likelihood) -ln(Likelihood) without constants -ln(Likelihood) without constants, σ known

Joint likelihood: Combining multiple data sets Share the parameter values for each data set Estimate the parameters while maximizing the combined likelihood (assuming independence) Think: Bernoulli → Binomial But, with the possibility of combining different likelihood functions

Using Likelihoods PARAMETER_SECTION. init_number sigma. PROCEDURE_SECTION pred_y=a+b*x; f=nobs*log(sigma) +0.5*sum(square((pred_y-y)/sigma));

.pin file #a 4 #b 2 #init_number sigma 1.5

Standard deviation file (*.std) index name value std dev 1 a e e-01 2 b e e-01 3 sigma e e-01

Correlation Matrix (*.COR) index name value std dev a4.0782e e b e e sigma e e